Answer:
- x-intercept: 2√2
- y-intercept: (2√2)/3
Step-by-step explanation:
You want the new x- and y-intercepts of the line 2x +y -2 = 0, after the axes are rotated 45° clockwise.
Rotation
Rotating the axes 45° clockwise is equivalent to rotating the line 45° CCW. The transformation that accomplishes that is ...
[tex](x,y)\Rightarrow\left(\dfrac{x+y}{\sqrt{2}},\dfrac{y-x}{\sqrt{2}}\right)[/tex]
This means the rotated line has equation ...
[tex]2\left(\dfrac{x+y}{\sqrt{2}}\right)+\left(\dfrac{y-x}{\sqrt{2}}\right)-2=0\\\\\\\dfrac{x}{2\sqrt{2}}+\dfrac{3y}{2\sqrt{2}}=1\qquad\text{collect terms, write in intercept form}[/tex]
In this last form of the equation, the x- and y-intercepts are the denominators of the x- and y-terms, respectively:
The x-intercept is 2√2.
The y-intercept is (2√2)/3.
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Additional comment
The attachment shows the original line (black) and the rotated axes (brown). The line rotated 45° CCW is shown in blue, and its intercepts are marked with their coordinates. Comparing the locations of the black line intercepts on the brown axes, you can see the correspondence to the marked intercept coordinates.
